Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,14,11)(4,5,10,9)(12,15,16,13)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(1,2,6,7)(3,11,14,8)(4,5,10,9)(12,13,16,15)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(34,40)(35,36)(42,45)(43,44);(2,7)(4,10)(8,11)(12,16)(19,24)(21,29)(27,30)(28,32)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,3)(2,8)(4,12)(5,15)(6,14)(7,11)(9,13)(10,16)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,4,6,10)(2,9,7,5)(3,12,14,16)(8,13,11,15)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,5,6,9)(2,10,7,4)(3,13,14,15)(8,12,11,16)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529099);